A company with a 12% WACC is offered two projects. One is a safe supermarket expansion. The other is a risky new technology venture. Both promise 13%. The WACC says accept both. But the investors who fund the company would want far more than 12% for the technology venture and could accept less for the supermarket.
The capital asset pricing model (CAPM) puts a price on risk so that each project is judged against a return that matches its own risk.
This topic covers systematic risk and the beta factor, the CAPM formula and the security market line, the market premium, the beta of a portfolio, alpha and abnormal returns, the evidence on CAPM and its strengths and weaknesses, the arbitrage pricing model, and the use of CAPM to find a project-specific discount rate (including an outline of ungearing and regearing betas).
The previous topic covered portfolio theory. The next covers sources of finance. The May 2014 and November 2014 Pathfinder papers both tested CAPM.
In 30 seconds
Beta (β) measures a security’s systematic risk against the market: β = ρsm × σs ÷ σm. The market has β = 1 and a risk-free asset β = 0. CAPM: Re = Rf + β(Rm − Rf), where Rm − Rf is the market premium. The formula is the equation of the security market line. A portfolio’s beta is the weighted average of its securities’ betas. Alpha is the actual return less the CAPM return.
CAPM assumes diversified investors, identical expectations, a perfect market and one period. Its advantage is that it prices only systematic risk. For projects with different business risk, use a project beta and the CAPM rate instead of the WACC. The arbitrage pricing model adds several risk factors.
Key terms in plain English
| Term | What it means |
|---|---|
| Beta factor | A security’s systematic risk relative to the market portfolio |
| Market premium | The market return less the risk-free return |
| Security market line | A line showing the required return for any beta |
| Alpha | The abnormal return above or below the CAPM return |
| Asset beta | The beta of a business with the effect of gearing removed |
| Equity beta | The beta of shares, including the effect of gearing |
| Arbitrage pricing model | A multi-factor model of required return |
Syllabus check
This topic is in syllabus area C (financing decisions). It covers:
- beta values as a measure of systematic risk;
- a required rate of return from the CAPM equation;
- the use of CAPM to find discount rates for projects.
CAMA 2020 and required returns
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria.
| Section | What it says | Link to this topic |
|---|---|---|
| s. 305 | A director must act in good faith in the best interests of the company | Projects should earn enough for their risk, for the benefit of the company |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should test the risk rate used to judge a project |
| s. 377(1) and s. 378 | Directors prepare statements that follow IFRS and give a true and fair view | Decisions later affect reported assets, profit and cash |
| s. 385 | The directors’ report covers performance and risk | Major plans and risks are reported to shareholders |
Systematic risk and beta
Investors with diversified portfolios care only about systematic risk, so the return they expect should reflect that alone. Every security except a risk-free one has some. It may be higher or lower than the market’s. If a security has more systematic risk, its return rises by more than the market’s in good times and falls by more in bad. The beta factor measures this against the market portfolio:
β = covariance of the security’s return with the market ÷ variance of the market = ρsmσs ÷ σm.
- The market portfolio has β = 1. A risk-free security has β = 0.
- A share with β above 1 is more volatile than the market. Below 1 is less volatile.
- Betas of listed companies are estimated by regression of past share returns on market returns and are published by data providers.
The CAPM formula
Re = Rf + β(Rm − Rf). The required return is the risk-free rate plus a premium for systematic risk: the market premium (Rm − Rf) times beta. ICAN’s example: Rf = 4%, Rm = 8.5%. For beta 1.25, R = 4% + 1.25 × 4.5% = 9.625%. For beta 0.90, R = 4% + 0.90 × 4.5% = 8.05%. The formula is the equation of the security market line (SML), which shows the required return for any beta. In the UK the market premium has been about 3.5% to 5%.
Portfolio beta
The beta of a portfolio is the weighted average of the betas of its securities. For weights 10%, 25%, 15%, 30% and 20% and betas 1.20, 0.90, 0.96, 1.15 and 1.06, the portfolio beta is 0.120 + 0.225 + 0.144 + 0.345 + 0.212 = 1.046.
Alpha and abnormal returns
If a security earns more or less than CAPM says, the difference is its alpha: R = Rf + β(Rm − Rf) + α. A positive alpha makes buyers want the share and pushes the price up; a negative alpha does the reverse. Alpha should be temporary and average zero for a diversified portfolio. ICAN’s example: the return is 11%, beta 1.10, risk-free 5%, market 8%. 11% = 5% + 1.10 × 3% + α, so α = 2.7%.
For bonds, a similar model uses “beta” = the bond’s duration ÷ the bond market’s duration and the redemption yield as the return. In ICAN’s example, Bond P (duration 5.0 against a market of 6.0) has a beta of 0.833, a required return of 6.0% against an actual 6.4%, a positive abnormal return of 0.4%.
Does CAPM predict well?
Fairly well. Shares with betas above 1 tend to return slightly less than CAPM says and those below 1 slightly more. It predicts badly in some industries, such as oil and gas and mining. It is probably the best model available.
Advantages and disadvantages
Assumptions: a perfect capital market with information available to all; identical investor expectations; investors are risk-averse and hold diversified portfolios; borrowing and lending at the risk-free rate; a single period.
Advantages: it uses only systematic risk, which matters to diversified investors; it is theoretically derived and widely tested; it is a better measure of the cost of equity than the dividend growth model; and it is better than WACC for project discount rates.
Disadvantages: it is hard to estimate the market premium, the risk-free rate (which maturity?) and beta (and project betas); it ignores unsystematic risk, which matters to undiversified owners and managers; it is a single-period model used on multi-period projects.
The arbitrage pricing model
CAPM has one factor. Other factors may affect returns: company size, unexpected changes in interest rates, industrial production or inflation. The arbitrage pricing model gives a premium for each: Re = Rf + β1(R1 − Rf) + β2(R2 − Rf) + …. Systematic risk is by far the main driver but probably not the only one.
CAPM and project appraisal
CAPM gives the cost of equity, which goes into the WACC. It is more reliable than the dividend growth model because it ignores unsystematic risk and uses betas measured from market data. Projects should be discounted at their marginal cost of capital, but usually the WACC is used on the assumption that one project will not change it.
That fails when a large project changes the capital structure or when the project has a different business risk (for example, car manufacturing is riskier than retailing; office construction may differ from housing; countries differ). In those cases, find a project beta and use CAPM to set a risk-adjusted rate, then calculate NPV at that rate.
Finding a project beta in outline
- Find a listed proxy company in the same business. Remove from its equity beta the effect of its gearing to get an asset beta. If it has other divisions, first strip those out.
- Re-gear the asset beta to the investing company’s target gearing to get an equity beta.
- Use CAPM for the cost of equity, combine with the post-tax cost of debt in a WACC at target gearing and discount the project at it.
ICAN’s November 2014 question: the proxy has an equity beta of 1.82, 60% of its value in cement and 40% in drinks with a drinks equity beta of 0.8. So the cement equity beta is x in 0.6x + 0.4 × 0.8 = 1.82, giving 2.5. After ungearing (debt to equity 40%, debt beta 0.4, tax 25%) the asset beta is about 2.0. Regearing to 30% debt to equity gives an equity beta of 2 + (2 − 0.4) × 0.3 × 0.75 = 2.36. With Rf 5% and a market premium 7%, ke = 5% + 2.36 × 7% = 21.52%.
The post-tax cost of debt is 8% × 0.75 = 6%, so WACC = (100 ÷ 130) × 21.52% + (30 ÷ 130) × 6% = about 18%, far above the company’s existing 11% because cement is riskier. The next topics on gearing explain the formulas in detail.
The rules in more depth
Why beta, not standard deviation
Standard deviation measures all risk. Beta measures only the part that cannot be diversified, which is the part investors are paid for.
Why a project can be rejected below the SML
If its IRR is below the CAPM return for its beta, it does not compensate shareholders for its systematic risk, even if the IRR beats the WACC.
Why asset betas are used for projects
An asset beta reflects business risk alone. Gearing adds financial risk, which has to be rebuilt for the investing company’s own gearing.
Why the premium is hard to estimate
Expected market returns are not observable, and past averages vary with the period chosen.
Effect on analysis
Compute beta, find the required return, compare it with the expected return or IRR and state the decision with the limits of the model.
Worked examples
Example 1: CAPM. The risk-free rate is 5% and the market return is 12%. Calculate the required return for betas of 1.3 and 0.8.
- Market premium = 7%.
- β 1.3: 5% + 1.3 × 7% = 14.1%.
- β 0.8: 5% + 0.8 × 7% = 10.6%.
Example 2: portfolio beta. A portfolio holds 40% of a share with β 1.4, 35% with β 0.9 and 25% with β 1.0.
- 0.40 × 1.4 = 0.560.
- 0.35 × 0.9 = 0.315
0.25 × 1.0 = 0.250. - Portfolio beta = 1.125.
Example 3: alpha. A share returns 14%. Its beta is 1.1, the risk-free rate is 5% and the market return is 11%.
- Required return.
5% + 1.1 × 6% = 11.6%. - Alpha = 14% − 11.6% = +2.4%.
- The share looks underpriced, so buying pressure should raise its price.
Example 4: a project below the line. A project has an IRR of 13% and a beta of 1.4. The company’s WACC is 12%. The risk-free rate is 5% and the market premium is 7%.
- Required return.
5% + 1.4 × 7% = 14.8%. - The IRR of 13% is below this, so the project does not compensate for its risk.
- Using the WACC of 12% would have wrongly accepted it.
How to score full marks
- Write the formula and show each input.
- Label the market premium separately from the market return.
- Compare the CAPM return with the actual or the IRR.
- Link beta to systematic risk and diversification.
- For projects, say why the WACC may be wrong.
- Give assumptions and limits.
Common mistakes in the exam
- Multiplying beta by the market return, not the premium.
- Forgetting to add the risk-free rate.
- Using the equity beta of a proxy without ungearing.
- Using the WACC for a project with different risk.
- Treating beta as total risk.
- Not stating the assumptions of CAPM.
Quick revision
- β = ρσs ÷ σm; market 1; risk-free 0.
- Re = Rf + β(Rm − Rf); this is the security market line.
- Portfolio beta is the weighted average.
- Alpha = actual − CAPM return.
- CAPM: single period, diversified investors, perfect market.
- Projects with different risk: project beta and a CAPM rate.
- Arbitrage pricing: several factors.
- CAMA 2020: ss. 305, 308, 377(1), 378, 385.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer. Full model answers follow the written questions.
Recall check: multiple choice
Recent ICAN papers are written, although older papers such as May 2014 included multiple-choice questions. These questions are for quick recall. The written questions that follow are closer to the exam.
Question 1. Beta measures:
- Systematic risk
- Total risk
- Unsystematic risk only
- Inflation
Show the answer
A. Relative to the market.
Question 2. The beta of the market portfolio is:
- 0
- 1
- 2
- Depends
Show the answer
B. The benchmark.
Question 3. The beta of a risk-free asset is:
- 0
- 1
- 0.5
- 2
Show the answer
A. It has no systematic risk.
Question 4. With R_f 15%, R_m 21% and beta 1.8, CAPM gives:
- 24.0%
- 25.8%
- 27.0%
- 36.0%
Show the answer
B. 15% + 1.8 × 6%.
Question 5. The CAPM equation is the equation of the:
- Security market line
- Capital market line
- Efficient frontier
- Yield curve
Show the answer
A. It plots required return against beta.
Question 6. Which is NOT a CAPM assumption?
- Investors have different expectations
- Investors are risk-averse
- A single period
- Borrowing and lending at the risk-free rate
Show the answer
A. Expectations are assumed identical.
Question 7. A portfolio beta is:
- The weighted average of the betas
- The highest beta
- The lowest beta
- Always 1
Show the answer
A. Weighted by proportions.
Question 8. Alpha is:
- Actual return minus the CAPM return
- The market return
- Beta squared
- The risk-free rate
Show the answer
A. An abnormal return.
Question 9. The arbitrage pricing model has:
- One factor
- Several risk factors
- No risk
- Only inflation
Show the answer
B. CAPM has one.
Question 10. Which section of CAMA 2020 sets the director’s duty of care and skill?
- Section 308
- Section 400
- Section 405
- Section 409
Show the answer
A. Directors should test the assumptions behind a decision.
Written questions
Question 11 (10 marks). (a) State four assumptions of CAPM. (4 marks) (b) State three advantages and three disadvantages of CAPM. (6 marks)
Question 12 (20 marks). The risk-free rate is 6% and the market return is 13%. Share X has a correlation with the market of 0.8 and a standard deviation of 30%; the market’s is 20%. Its expected return is 16%. Share Y has a beta of 0.9 and an expected return of 11%. (a) Calculate beta for X. (3 marks) (b) Calculate the required returns and alphas of X and Y and say which is under- or over-priced.
(7 marks) (c) Calculate the beta of a portfolio that is half X and half Y. (2 marks) (d) A project has an IRR of 15% and a beta of 1.3. The WACC is 12%. Advise. (5 marks) (e) State one limit of CAPM. (3 marks)
Question 13 (10 marks). (a) Explain how CAPM may be used to find a project-specific discount rate. (6 marks) (b) Explain how the arbitrage pricing model differs from CAPM. (4 marks)
Model answers
Show the model answer to Question 11
(a) Assumptions (4 marks). A perfect capital market with all information available; identical expectations; investors hold diversified portfolios and are risk-averse; borrowing and lending at the risk-free rate; a single period. Any four. (1 each)
(b) Advantages and disadvantages (6 marks). Advantages: considers only systematic risk; theoretically based and tested; better than the dividend growth model; better than WACC for project rates. Disadvantages: the market premium, risk-free rate and beta are hard to estimate; it ignores unsystematic risk; it is a single-period model. Any three of each. (1 each)
Show the model answer to Question 12
(a) Beta of X (3 marks). β = 0.8 × 30 ÷ 20 = 1.2. (3)
(b) Required returns (7 marks).
- Market premium = 7%.
- X: 6% + 1.2 × 7% = 14.4%
- alpha = 16% − 14.4% = +1.6%, so underpriced. (3.5)
- Y: 6% + 0.9 × 7% = 12.3%
- alpha = 11% − 12.3% = −1.3%, so overpriced. (3.5)
(c) Portfolio beta (2 marks). 0.5 × 1.2 + 0.5 × 0.9 = 1.05. (2)
(d) Project (5 marks).
- Required return.
6% + 1.3 × 7% = 15.1%. (2) - The IRR of 15% is below it, so the project does not compensate for its systematic risk and should be rejected. (2)
- The WACC of 12% would wrongly have accepted it. (1)
(e) Limit (3 marks). The inputs (market premium, risk-free rate and beta) are hard to estimate, or CAPM is a single-period model used on multi-year projects. (3)
Show the model answer to Question 13
(a) Project rate (6 marks). The project’s business risk may differ from the company’s, so the WACC is wrong. (1.5) Find a listed proxy in the same business, ungear its equity beta to get an asset beta. (1.5) Regear to the company’s target gearing. (1.5) Use CAPM for the cost of equity, combine with debt in a WACC and discount the project at it. (1.5)
(b) APM (4 marks). CAPM has one factor, systematic market risk. (2) The arbitrage pricing model has several factors, such as size, inflation, interest rates and industrial production, each with its own beta and premium. (2)
Pathfinder on Beta, CAPM, alpha and project-specific discount rates
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Multiple-choice 6 asked for the name of the line linking expected return to beta (the security market line), 7 asked for a CAPM cost of equity (25.8%) and 14 asked which statement is NOT a CAPM assumption (that investors have different expectations). Short answers 12 and 15 asked for the capital asset pricing model and the arbitrage pricing theory. Read the sections on the CAPM formula and the arbitrage pricing model.
Revise first: The CAPM formula
The compulsory question asked for the cost of capital for a cement project, using a proxy company’s beta, ungearing and regearing and CAPM. The result is about 18% against the company’s 11%, because cement is riskier. Read the section on CAPM and project appraisal.
Revise first: CAPM and project appraisal
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: May 2014 multiple-choice 6, 7 and 14 and short-answer 12 and 15; November 2014 question 1(a).
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